The Multiscale Perturbation Method For Two Phase Flows In Porous Media
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The Multiscale Perturbation Method for Two-phase Flows in Porous Media
This dissertation is focused on creating new multiscale mixed methods that can help reduce the computational cost and improve the accuracy of the numerical solution to the two-phase, immiscible, incompressible flow problem obtained by the operator splitting technique. The two-phase flow governing systems of equations consists of two partial differential equations:(i) a second order elliptic equation or a Poisson equation, (ii) a hyperbolic conservation law. Typically in an operator splitting technique the elliptic and the hyperbolic equations are solved sequentially. The procedure to solve the elliptic equation numerically is computationally very expensive once the size of the linear system to be solved is large. Considering that, this dissertation focuses on the development of fast and efficient solvers which are naturally parallelizable to compute the solution of the second order elliptic equation that is approximated numerically. At first, the Multiscale Perturbation Method for second order elliptic equations (MPM) is presented . This method is based on the Multiscale Mixed Method (MuMM). The MuMM, like most multiscale methods solves the elliptic equation numerically by first computing a set of local multiscale mixed basis functions (MMBFs) with special boundary condition (Robin in this case) and then solves the global problem. The MPM proposes a new algorithm that can reuse the MMBFs computed at an initial time as well as take advantage of a good initial guess by using classical perturbation techniques. Secondly, the MPM is then incorporated into the operator splitting algorithm to create a new modified algorithm (MPM-2P:Multiscale Perturbation Method for two-phase flows) that solves the two-phase flow equations numerically. A relative cost reduction which is the gain in terms of the computational time is computed between the solution obtained via the new MPM-2P and by using simply the MuMM. The results show an exceptional speed up - a reduction in computational cost from 60.8% to 96.7% - for the elliptic equation of realistic reservoir flow problems indicating that a large number of MuMM solutions in a traditional operator splitting method can be easily replaced by the inexpensive MPM-2P solutions. This makes it the most important contribution of this dissertation. Finally, a new multiscale mixed method with overlapping domain decomposition (O-MuMM) is formulated and a parallel algorithm that implements the O-MuMM computationally is proposed. Numerical results for all three methods are discussed and analyzed to prove the validity of the new methods proposed.
Multiscale Finite Element Methods
Author: Yalchin Efendiev
language: en
Publisher: Springer Science & Business Media
Release Date: 2009-01-10
The aim of this monograph is to describe the main concepts and recent - vances in multiscale ?nite element methods. This monograph is intended for thebroaderaudienceincludingengineers,appliedscientists,andforthosewho are interested in multiscale simulations. The book is intended for graduate students in applied mathematics and those interested in multiscale compu- tions. It combines a practical introduction, numerical results, and analysis of multiscale ?nite element methods. Due to the page limitation, the material has been condensed. Each chapter of the book starts with an introduction and description of the proposed methods and motivating examples. Some new techniques are introduced using formal arguments that are justi?ed later in the last chapter. Numerical examples demonstrating the signi?cance of the proposed methods are presented in each chapter following the description of the methods. In the last chapter, we analyze a few representative cases with the objective of demonstrating the main error sources and the convergence of the proposed methods. A brief outline of the book is as follows. The ?rst chapter gives a general introductiontomultiscalemethodsandanoutlineofeachchapter.Thesecond chapter discusses the main idea of the multiscale ?nite element method and its extensions. This chapter also gives an overview of multiscale ?nite element methods and other related methods. The third chapter discusses the ext- sion of multiscale ?nite element methods to nonlinear problems. The fourth chapter focuses on multiscale methods that use limited global information.
Domain Decomposition Methods in Science and Engineering XVII
Author: Ulrich Langer
language: en
Publisher: Springer Science & Business Media
Release Date: 2008-01-02
Domain decomposition is an active, interdisciplinary research field concerned with the development, analysis, and implementation of coupling and decoupling strategies in mathematical and computational models. This volume contains selected papers presented at the 17th International Conference on Domain Decomposition Methods in Science and Engineering. It presents the newest domain decomposition techniques and examines their use in the modeling and simulation of complex problems.